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This paper studies solutions of the functional equation \ f (x) = ₍ = ₀N cₙ f (kx - n), \ where k 2 is an integer, and ₍ = ₀N cₙ = k. Part I showed that equations of this type have at most one L¹ -solution up to a multiplicative constant, which necessarily has compact support in 0, N / {k - 1}. This paper gives a time-domain representation for such a function f (x) (if it exists) in terms of infinite products of matrices (that vary as x varies). Sufficient conditions are given on \ {cₙ \} for a continuous nonzero L¹ -solution to exist. Additional conditions sufficient to guarantee f Cʳ are also given. The infinite matrix product representations is used to bound from below the degree of regularity of such an L¹ -solution and to estimate the Hölder exponent of continuity of the highest-order well-defined derivative of f (x). Such solutions f (x) are often smoother at some points than others. For certain f (x) a hierarchy of fractal sets in R corresponding to different Hölder exponents of continuity for f (x) is described.
Daubechies et al. (Wed,) studied this question.